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NAME CGGHRD - reduce a pair of complex matrices (A,B) to general- ized upper Hessenberg form using unitary similarity transformations, where A is a (generally non-symmetric) square matrix and B is upper triangular SYNOPSIS SUBROUTINE CGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO ) CHARACTER COMPQ, COMPZ INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ), Z( LDZ, * ) PURPOSE CGGHRD reduces a pair of complex matrices (A,B) to general- ized upper Hessenberg form using unitary similarity transformations, where A is a (generally non-symmetric) square matrix and B is upper triangular. More precisely, CGGHRD simultaneously decomposes A into Q H Z* and B into Q T Z* , where H is upper Hessenberg, T is upper tri- angular, Q and Z are unitary, and * means conjugate tran- spose. If COMPQ and COMPZ are 'V' or 'I', then the unitary transformations used to reduce (A,B) are accumulated into the arrays Q and Z s.t.: Q(in) A(in) Z(in)* = Q(out) A(out) Z(out)* Q(in) B(in) Z(in)* = Q(out) B(out) Z(out)* CGGHRD implements an unblocked form of the method, there being no blocked form known at this time. ARGUMENTS COMPQ (input) CHARACTER*1 = 'N': do not compute Q; = 'V': accumulate the row transformations into Q; = 'I': overwrite the array Q with the row transfor- mations. COMPZ (input) CHARACTER*1 = 'N': do not compute Z; = 'V': accumulate the column transformations into Z; = 'I': overwrite the array Z with the column transformations. N (input) INTEGER The number of rows and columns in the matrices A, B, Q, and Z. N >= 0. ILO (input) INTEGER IHI (input) INTEGER It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO >= 1 and IHI <= N. A (input/output) COMPLEX array, dimension (LDA, N) On entry, the N-by-N general matrix to be reduced. On exit, the upper triangle and the first subdiago- nal of A are overwritten with the Hessenberg matrix H, and the rest is set to zero. LDA (input) INTEGER The leading dimension of the array A. LDA >= max( 1, N ). B (input/output) COMPLEX array, dimension (LDB, N) On entry, the N-by-N upper triangular matrix B. On exit, the upper triangular matrix T = Q* B Z. The entries below the diagonal are set to zero. LDB (input) INTEGER The leading dimension of the array B. LDB >= max( 1, N ). Q (input/output) COMPLEX array, dimension (LDQ, N) If COMPQ='N', then Q will not be referenced. If COMPQ='V', then the conjugate transpose of the Givens transformations which are applied to A and B on the left will be applied to the array Q on the right. If COMPQ='I', then Q will first be overwrit- ten with an identity matrix, and then the Givens transformations will be applied to Q as in the case COMPQ='V'. LDQ (input) INTEGER The leading dimension of the array Q. LDQ >= 1. If COMPQ='V' or 'I', then LDQ >= N. Z (input/output) COMPLEX array, dimension (LDZ, N) If COMPZ='N', then Z will not be referenced. If COMPZ='V', then the Givens transformations which are applied to A and B on the right will be applied to the array Z on the right. If COMPZ='I', then Z will first be overwritten with an identity matrix, and then the Givens transformations will be applied to Z as in the case COMPZ='V'. LDZ (input) INTEGER The leading dimension of the array Z. LDZ >= 1. If COMPZ='V' or 'I', then LDZ >= N. INFO (output) INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. FURTHER DETAILS This routine reduces A to Hessenberg and B to triangular form by an unblocked reduction, as described in _Matrix_Computations_, by Golub and van Loan (Johns Hopkins Press).